Here we show that if some zn is farther than 2 from the origin, then successive iterates will grow without bound. That is, they will run away to ∞. |
For a complex number zn = xn + i⋅yn, the absolute value is |
|zn| = √(xn2 + yn2), |
the distance from zn to the origin. |
Recalling the sequence z0, z1, ... is defined
by |
So suppose |
Because |zn| > 2, we can write |
|zn| = 2 + ε, |
for some |
Now |
|zn2| = |zn2 + c - c| ≤ |zn2 + c| + |c| |
So |
|zn2 + c| ≥ |zn2| - |c| = |zn|2 - |c| |
> |zn|2 - |zn| (because |zn| > |c|) |
= (|zn| - 1)⋅|zn| = (1 + ε)⋅|zn| |
That is, |zn+1| > (1 + ε)⋅|zn|. Iterating,
|
To complete the proof that |zn| > 2 implies the sequence runs
away to infinity, observe that if |
z0 = 0 |
z1 = c |
and z2 = c2 + c = c⋅(c + 1) |
so |z2| = |c|⋅|c + 1| > |c| (noting |c + 1| > 1 because |c| > 2). |
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